I. Vectors and Geometry in Two and Three Dimensions


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1 I. Vectors and Geometry in Two and Three Dimensions I.1 Points and Vectors Each point in two dimensions may be labeled by two coordinates (a,b) which specify the position of the point in some units with respect to some axes as in the figure on the left below. Similarly, each point in three dimensions may be labeled by three coordinates (a,b,c). The set of all points in two dimensions is y a (a,b) b x x a z b (a,b,c) denoted IR 2 and the set of all points is three dimensions is denoted IR 3. The distance from the point (x,y,z) to the point (x,y,z ) is (x x ) 2 +(y y ) 2 +(z z ) 2 so that the equation of the sphere centered on (1,2,3) with radius 4 is (x 1) 2 +(y 2) 2 +(z 3) 2 = 16. A vector is a quantity which has both a direction and a magnitude, like a velocity or a force. To specify a vector in three dimensions you have to give three components, just as for a point. To draw the vector with components a, b, c you can draw an arrow from the point (0,0,0) to the point (a,b,c). Similarly, to y (a,b) b a z c y (a,b,c) c y a x x b specify a vector in two dimensions you have to give two components and to draw the vector with components a, b you can draw an arrow from the point (0,0) to the point (a,b). There are many situations in which it is preferable to draw a vector with its tail at some point other than the origin. For example, suppose that you are analyzing the motion of a pendulum. τ r There are three forces acting on the pendulum bob: gravity g, which is pulling the bob straight down, tension τ in the rod, which is pulling the bob in the direction of the rod, and air resistance r, which is pulling the c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 1 g
2 bob in a direction opposite to its direction of motion. All three forces are acting on the bob. So it is natural to draw all three arrows representing the forces with their tails at the ball. To distinguish between the components of a vector and the coordinates of the point at its head, when its tail is at some point other than the origin, we shall use square rather than round brackets around the components of a vector. For example, here is the two dimensional vector [2, 1] drawn in three different positions. In each case, when the tail is at the point (u,v) the head is at (2 + u,1 + v). We warn you that, out in the real world, no one uses notation that distinguishes between components of a vector and the coordinates of its head. It is up to you to keep straight which is being referred to. y [2,1] (6,3) (2,1) (4,2) [2,1] (10, 1) (0,0) (8,0) x Exercises for I.1 1) Describe and sketch the set of all points (x,y) in IR 2 that satisfy a) x = y b) x+y = 1 c) x 2 +y 2 = 4 d) x 2 +y 2 = 2y 2) Describe and sketch the set of all points (x,y,z) in IR 3 that satisfy a) z = x b) x+y +z = 1 c) x 2 +y 2 +z 2 = 4 d) x 2 +y 2 +z 2 = 4, z = 1 e) x 2 +y 2 = 4 f) z > x 2 +y 2 3) The pressure p(x,y) at the point (x,y) is determined by x 2 2px+y 2 +1 = 0. Sketch several isobars. An isobar is a curve with equation p(x,y) = c for some constant c. 4) Consider any triangle. Pick a coordinate system so that one vertex is at the origin and a second vertex is on the positive x axis. Call the coordinates of the second vertex (a,0) and those of the third vertex (b, c). Find the circumscribing circle (the circle that goes through all three vertices). I.2 Addition of Vectors and Multiplication of a Vector by a Number These two operations have the obvious definitions a = [a 1,a 2 ], b = [b 1,b 2 ] = a+ b = [a 1 +b 1,a 2 +b 2 ] a = [a 1,a 2 ], s a number = s a = [sa 1,sa 2 ] and similarly in three dimensions. Pictorially, you add b to a by drawing b starting at the head of a and then drawing a vector from the tail of a to the head of b. To draw s a, you just change a s length by the (signed) factor s. a 2 +b 2 b 2 a 2 a+ b b 2a 2 a 2 a 2 a a a 2 a c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 2
3 These operations rarely cause any problems, because they inherit from the real numbers the properties of addition and multiplication that you are used to. Using 0 to denote the vectorall of whose components are zero and a to denote the vector each of whose components is the negative of the corresponding component of a (so that [a 1,a 2 ] = [ a 1, a 2 ]) 1. a+ b = b+ a 2. a+( b+ c) = ( a+ b)+ c 3. a+ 0 = a 4. a+( a) = 0 5. s( a+ b) = s a+s b 6. (s+t) a = s a+t a 7. (st) a = s(t a) 8. 1 a = a To subtract b from a pictorially, you may add b (which is drawn by reversing the direction of b) to a. Alternatively, if you draw a and b with their tails at a common point, then a b is the vector from the head of b to the head of a. That is, a b is the vector you must add to b in order to get a. b a b b a a b There are some vectors that occur sufficiently commonly that they are given special names. One is the vector 0. Some others are the standard basis vectors in two dimensions y z î = [1,0] ĵ = [0,1] ĵ î ˆk and the standard basis vectors in three dimensions x ĵ y î = [1,0,0] ĵ = [0,1,0] ˆk = [0,0,1] î x Some people rename î, ĵ and ˆk to ê 1, ê 2 and ê 3 respectively. Using the above properties we have, for all vectors, [a 1,a 2 ] = a 1 î+a 2 ĵ [a 1,a 2,a 3 ] = a 1 î+a 2 ĵ+a 3ˆk A sum of numbers times vectors, like a 1 î+a 2 ĵ is called a linear combination of the vectors. Thus all vectors can be expressed as linear combinations of the standard basis vectors. The hatsˆare used to signify that the standard basis vectors are unit vectors, meaning that they are of length one, where the length of a vector is defined by a = [a 1,a 2 ] = a = a 2 1 +a2 2 a = [a 1,a 2,a 3 ] = a = a 2 1 +a2 2 +a2 3 Exercises for I.2 1) Let a = [2,0] and b = [1,1]. Evaluate and sketch a+ b, a+2 b and 2 a b. 2) Find the equation of a sphere if one of its diameters has end points (2,1,4) and (4,3,10). 3) Determine whether or not the given points are collinear (that is, lie on a common straight line) a) (1,2,3), (0,3,7), (3,5,11) b) (0,3, 5), (1,2, 2), (3,0,4) 4) Show that the set of all points P that are twice as far from (3, 2,3) as from (3/2,1,0) is a sphere. Find its centre and radius. 5) Show that the diagonals of a parallelogram bisect each other. c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 3
4 I.3 The Dot Product There are three types of products used with vectors. The first is multiplication by a scalar, which we have already seen. The second is the dot product, which is defined by a = [a 1,a 2 ], b = [b1,b 2 ] = a b = a 1 b 1 +a 2 b 2 a = [a 1,a 2,a 3 ], b = [b 1,b 2,b 3 ] = a b = a 1 b 1 +a 2 b 2 +a 3 b 3 in two and three dimensions respectively. The properties of the dot product are as follows: 0. a, b are vectors and a b is a number 1. a a = a 2 2. a b = b a 3. a ( b+ c) = a b+ a c, ( a+ b) c = a c+ b c 4. (s a) b = s( a b) 5. 0 a = 0 6. a b = a b cosθ where θ is the angle between a and b 7. a b = 0 a = 0 or b = 0 or a b Properties 0 through 5 are almost immediate consequences of the definition. For example, for property 3 in dimension 2, a ( b+ c) = [a 1,a 2 ] [b 1 +c 1,b 2 +c 2 ] = a 1 (b 1 +c 1 )+a 2 (b 2 +c 2 ) = a 1 b 1 +a 1 c 1 +a 2 b 2 +a 2 c 2 a b+ a c = [a 1,a 2 ] [b 1,b 2 ]+[a 1,a 2 ] [c 1,c 2 ] = a 1 b 1 +a 2 b 2 +a 1 c 1 +a 2 c 2 Property 6 is sufficiently important that it is often used as the definition of dot product. It is not at all an obvious consequence of the definition. To verify it, we just write a b 2 in two different ways. The first expresses a b 2 in terms of a b. It is a b 2 1 =( a b) ( a b) 3 = a a a b b a+ b b 1,2 = a 2 + b 2 2 a b 1 Here, =, for example, means that the equality is a consequence of property 1. The second way we write a b 2 involves cosθ and follows from the cosine law. Just in case you don t remember the cosine law, we prove it along the way. From the figure we have a a b a a b a sinθ θ b b a cosθ b a cosθ a b 2 = ( b a cosθ ) 2 ( ) 2 + a sinθ = b 2 2 a b cosθ + a 2 cos 2 θ + a 2 sin 2 θ = b 2 2 a b cosθ + a 2 c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 4
5 Setting the two expressions for a b 2 equal to each other, a b 2 = a 2 + b 2 2 a b = b 2 2 a b cosθ + a 2 cancelling the a 2 and b 2 common to both expressions and dividing by 2 gives 2 a b = 2 a b cosθ a b = a b cosθ which is property 6. Property 7 follows directly from property 6: a b = a b cosθ is zero if and only if at least one of the three factors a, b, cosθ is zero. The first factor is zero if and only if a = 0. The second factor is zero if and only if b = 0. The third factor is zero if and only if θ = ± π 2 +2kπ, for some integer k, which in turn is true if and only if a and b are mutually perpendicular. Because of Property 7, the dot product can be used to test whether or not two vectors are orthogonal. Orthogonal is just another name for perpendicular. Testing for orthogonality is one of the main uses of the dot product. Another is computing projections. Draw two vectors, a and b, with their tails at a common point and drop a perpendicular from the head of a to the line that passes through both the head and tail of b. By definition, the projection of the vector a on the vector b is the vector from the tail of b to the point on the line where the perpendicular hits. θ a b proj b a a θ proj b a b Let θ be the angle between a and b. If θ is no more than 90, as in the figure on the left above, the length of the projection of a on b is a cosθ. By property 6, a cosθ = a b/ b, so the projection is a vector whose length is a b/ b and whose direction is given by the unit vector b/ b. Hence projection of a on b = proj b a = a b b b b = a b b b 2 If θ is larger than 90, as in the figure on the right above, the projection has length a cos(π θ) = a cosθ = a b/ b and direction b/ b. In this case proj b a = a b b b b = a b b b 2 So the formula proj b a = a b b is applicable whenever b 0. One use of projections is to resolve forces. b 2 There is an example in the next section. Exercises for I.3 1) Compute the dot product of the vectors a and b. Find the angle between them. a) a = (1,2), b = ( 2,3) b) a = ( 1,1), b = (1,1) c) a = (1,1), b = (2,2) d) a = (1,2,1), b = ( 1,1,1) e) a = ( 1,2,3), b = (3,0,1) c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 5
6 2) Let a = [a 1,a 2 ]. Compute the projection of a on î and ĵ. 3) Does the triangle with vertices (1,2,3), (4,0,5) and (3,6,4) have a right angle? 4) Let O = (0,0), A = (a,0) and B = (b,c) be the three vertices of the triangle in problem 4 of I.1. Let U be the centre of the circle through O, A and B. Guess proj OA OU and proj OB OU. Compute proj OA OU and proj OB OU. I.4 Application of Dot Products to Resolution of Forces The Pendulum Model a pendulum by a mass m that is connected to a hinge by an idealized rod that is massless and of fixed length l. Denote by θ the angle between the rod and vertical. The forces acting on the βl dθ dt θ l τ mass are gravity, which has magnitude mg and direction (0, 1), tension in the rod, whose magnitude τ(t) automatically adjusts itself so that the distance between the mass and the hinge is fixed at l and whose direction is always parallel to the rod and possibly some frictional forces, like friction in the hinge and air resistance. Assume that the total frictional force has magnitude proportional to the speed of the mass and has direction opposite to the direction of motion of the mass. If we use a coordinate system centered on the hinge, the (x,y) coordinates of the mass at time t are mg x(t) = lsinθ(t) y(t) = lcosθ(t) where θ(t) is the angle between the rod and vertical at time t. So, the velocity and acceleration vectors of the mass are v(t) = d dt [x(t),y(t)] = l[d dt sinθ(t), d dθ dt cosθ(t)] = l[cosθ(t),sinθ(t)] dt (t) { [cosθ(t),sinθ(t)] dθ dt (t)} = l[cosθ(t),sinθ(t)] d2 θ dt (t)+l[ d 2 dt cosθ(t), d dθ dt sinθ(t)] dt (t) a(t) = d2 dt [x(t),y(t)] = l d 2 dt = l[cosθ(t),sinθ(t)] d2 θ dt (t)+l[ sinθ(t),cosθ(t)] ( dθ 2 dt (t)) 2 The negative of the velocity vector is l[cosθ,sinθ] dθ dt, so the total frictional force is βl[cosθ,sinθ]dθ dt for some constant of proportionality β. The vector τ(t)[ sinθ(t),cosθ(t)] has magnitude τ(t) and direction parallel to the rod pointing from the mass towards the hinge and so is the force due to tension in the rod. Hence, for this physical system, Newton s law of motion is mass acceleration = applied force ml[cosθ,sinθ] d2 θ dt +ml[ sinθ,cosθ] ( ) dθ 2 2 dt = mg[0, 1]+τ[ sinθ,cosθ] βl[cosθ,sinθ] dθ dt This rather complicated equation can be considerably simplified (and consequently better understood) by taking its components parallel to and perpendicular to the direction of motion. From the velocity vector v(t), we see that [cosθ(t),sinθ(t)] is a unit vector parallel to the direction of motion at time t. In general, the projection of any vector b on any unit vector ˆd is b ˆd ˆd ˆd 2 = ( b ˆd) ˆd c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 6 (I.1)
7 The coefficient b ˆd is, by definition, the component of b in the direction ˆd. So, by dotting both sides of the equation of motion (I.1) with ˆd = [cosθ(t),sinθ(t)], we extract the component parallel to the direction of motion. Since [cosθ,sinθ] [cosθ,sinθ] = 1 this gives [cosθ,sinθ] [ sinθ,cosθ] = 0 [cosθ,sinθ] [0, 1] = sinθ ml d2 θ dt = mgsinθ βl dθ 2 dt When θ is small, we can approximate sinθ θ and get the equation d 2 θ dt + β dθ 2 m dt + g l θ = 0 which is easily solved. In 4, we shall develop an algorithm for finding the solution. For now, we ll just guess. When there is no friction (so that β = 0), we would expect the pendulum to just oscillate. So it is natural to guess θ(t) = A sin(ωt δ), which is an oscillation with (unknown) amplitude A, frequency ω (radians per unit time) and phase δ. Substituting the guess into the left hand side θ + g l θ yields Aω2 sin(ωt δ)+a g l sin(ωt δ), which is zero if ω = g/l. So θ(t) = Asin(ωt δ) is a solution for any amplitude A and phase δ provided the frequency ω = g/l. When there is some, but not too much, friction, so that β > 0 is relatively small, we would expect oscillation with decaying amplitude. So we guess θ(t) = Ae γt sin(ωt δ). With this guess, θ(t) = Ae γt sin(ωt δ) θ (t) = γae γt sin(ωt δ)+ ωae γt cos(ωt δ) θ (t) = (γ 2 ω 2 )Ae γt sin(ωt δ) 2γωAe γt cos(ωt δ) and the left hand side ] ] d 2 θ dt + β dθ 2 m dt + g l [γ θ = 2 ω 2 β m γ + g l Ae γt sin(ωt δ)+ [ 2γω+ βm ω Ae γt cos(ωt δ) vanishes if γ 2 ω 2 β m γ + g l = 0 and 2γω+ β β mω = 0. The second equation tells us the decay rate γ = 2m and then the first tells us the frequency ω = γ 2 βm γ + gl = g l β2 4m 2 When there is a lot of friction (namely when β2 4m > g 2 l, so that the frequency ω is not a real number), we would expect damping without oscillation and so would guess θ(t) = Ae γt. To extract the components perpendicular to the direction of motion, we dot with [ sinθ,cosθ] rather than [cosθ,sinθ]. Note that, because [ sinθ,cosθ] [cosθ,sinθ] = 0, [ sinθ,cosθ] really is perpendicular to the direction of motion. Since [ sinθ,cosθ] [cosθ,sinθ] = 0 [ sinθ,cosθ] [ sinθ,cosθ] = 1 [ sinθ,cosθ] [0, 1] = cosθ dotting both sides of the equation of motion (I.1) with [ sinθ,cosθ] gives ml ( dθ dt) 2 = mgcosθ +τ This equation just determines the tension τ = ml ( dθ dt) 2 +mgcosθ in the rod, once you know θ(t). c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 7
8 Exercises for I.4 1) Consider a skier who is sliding without friction on the hill y = h(x) in a two dimensional world. The skier is subject to two forces. One is gravity. The other acts perpendicularly to the hill. The second force automatically adjusts its magnitude so as to prevent the skier from burrowing into the hill. Suppose that the skier became airborne at some (x 0,y 0 ) with y 0 = h(x 0 ). How fast was the skier going? 2) A marble is placed on the plane ax+by +cz = d. The coordinate system has been chosen so that the positive z axis points straight up. The coefficient c is nonzero and the coefficients a and b are not both zero. In which direction does the marble roll? Why were the conditions c 0 and a,b not both zero imposed? I.5 Areas of Parallelograms Construct a parallelogram as follows. Pick two vectors [a,b] and [c,d]. Draw them with their tails at a common point. Then draw [a,b] a second time with its tail at the head of [c,d] and draw [c,d] a second time with its tail at the head of [a,b]. If the the common point is the origin, you get a picture like the figure below. Any parallelogram can be constructed like this if you pick the common point and two vectors c a (a+c,b+d) b d (c,d) (a,b) d b a appropriately. Let s compute the area of the parallelogram. The area of the large rectangle with vertices (0,0), (0,b+d), (a+c,0) and (a+c,b+d) is (a+c)(b+d). The parallelogramwe want can be extracted from the large rectangle by deleting the two small rectangles (each of area bc) the two lightly shaded triangles (each of area 1 2 cd) and the two darkly shaded triangles (each of area 1 2ab). So the desired area = (a+c)(b+d) 2 bc cd 2 1 2ab = ad bc In the above figure, we have implicitly assumed that a, b, c, d 0 and d/c b/a. In words, we have assumed that both vectors [a,b], [c,d] lie in the first quadrant and that [c,d] lies above [a,b]. By simply interchanging a c and b d in the picture and throughout the argument, we see that when a, b, c, d 0 and b/a d/c, so that the vector [c,d] lies below [a,b], the area of the parallelogram is bc ad. In fact, all cases are covered by the formula area of parallelogram with sides [a,b] and [c,d] = ad bc c Given two vectors [a,b] and [c,d], the expression ad bc is generally written det [ ] a b = ad bc c d and is called the determinant of the matrix [ ] a b c d c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 8
9 with rows [a,b] and [c,d]. The determinant of a 2 2 matrix is the product of the diagonal entries minus the product of the off diagonal entries. There is a similar formula in three dimensions. Any three vectors a = [a 1,a 2,a 3 ], b = [b 1,b 2,b 3 ] and c = [c 1,c 2,c 3 ] in three dimensions determine a parallelopiped (three a b c dimensional parallelogram). Its volume is given by the formula volume of parallelopiped with edges a, b, c = det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 The determinant of a 3 3 matrix can be defined in terms of some 2 2 determinants by det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 a 2 det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 +a 3 det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 (b 2 c 3 b 3 c 2 ) a 2 (b 1 c 3 b 3 c 1 ) +a 3 (b 1 c 2 b 2 c 1 ) This formula is called expansion along the top row. There is one term in the formula for each entry in the top row of the 3 3 matrix. The term is a sign times the entry itself times the determinant of the 2 2 matrix gotten by deleting the row and column that contains the entry. The sign alternates, starting with a +. We shall not prove this formula completely. But, there is one case in which we can easily verify that the volume of the parallelopiped is really given by the absolute value of the claimed determinant. If the vectors b and c happen to lie in the xy plane, so that b3 = c 3 = 0, then det a 1 a 2 a 3 b 1 b 2 0 = a 1 (b 2 0 0c 2 ) a 2 (b 1 0 0c 1 )+a 3 (b 1 c 2 b 2 c 1 ) c 1 c 2 0 = a 3 (b 1 c 2 b 2 c 1 ) The first factor, a 3, is the z coordinate of the one vector not contained in the xy plane. It is (up to a sign) the height of the parallelopiped. The second factor is, up to a sign, the area of the parallelogram determined by b and c. This parallelogram forms the base of the parallelopiped. The product is indeed, up to a sign, the volume of the parallelopiped. That the formula is true in general is a consequence of the fact (that we will not prove) that the value of a determinant does not change when one rotates the coordinate system and that one can always rotate our coordinate axes around so that b and c both lie in the xy plane. Exercises for I.5 1) Derive the formula area of parallegram = ad bc in the case when (a,b) lies in the first quadrant and (c,d) lies in the second quadrant. 2 a) Let [a,b] be a vector. Let r be the length of [a,b] and θ the angle between [a,b] and the x axis. Express a and b in terms of r and θ. b) Let [A,B] be the vector gotten by rotating [a,b] by an angle ϕ about its tail. Express A and B in terms of a, b and ϕ. 3) Let [a,b] and [c,d] be two vectors. Let [A,B] be the vector gotten by rotating [a,b] by an angle ϕ about its tail. Let [C,D] be the vector gotten by rotating [c,d] by the same angle ϕ about its tail. Show that [ ] [ ] a b A B det = det c d C D c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 9
10 I.6 The Cross Product We have already seen two different products involving vectors multiplication by scalars and the dot product. There is a third product, called the cross product that is defined by a = [a 1,a 2,a 3 ], b = [b 1,b 2,b 3 ] = a b = [a 2 b 3 a 3 b 2,a 3 b 1 a 1 b 3,a 1 b 2 a 2 b 1 ] Note that each component has the form a i b j a j b i. The index i of the first a in component number k of a b is just after k in the list 1,2,3,1,2,3,1,2,3,. The index j of the first b is just before k in the list. ( a b) k = a just after k b just before k a just before k b just after k For example, for component number k = 3, } just after 3 = 1 just before 3 = 2 = ( a b) 3 = a 1 b 2 a 2 b 1 There is a much better way to remember this definition. Recall that a 2 2 matrix is an array of numbers having two rows and two columns and that the determinant of a 2 2 matrix is the product of the entries on the diagonal minus the product of the entries not on the diagonal. A 3 3 matrix is an array of numbers having three rows and three columns. i j k a 1 a 2 a 3 b 1 b 2 b 3 You will shortly see why I have given the matrix entries rather peculiar names. The determinant of a 3 3 matrix can be defined in terms of some 2 2 determinants by det i j k a 1 a 2 a 3 = i det i j k a 1 a 2 a 3 jdet i j k a 1 a 2 a 3 +kdet i j k a 1 a 2 a 3 b 1 b 2 b 3 b 1 b 2 b 3 b 1 b 2 b 3 b 1 b 2 b 3 = i(a 2 b 3 a 3 b 2 ) j(a 1 b 3 a 3 b 1 ) +k(a 1 b 2 a 2 b 1 ) This formula is called expansion along the top row. There is one term in the formula for each entry in the top row. The term is a sign times the entry itself times the determinant of the 2 2 matrix gotten by deleting the row and column that contains the entry. The sign alternates, starting with a +. The formula for a b is gotten by replacing i by î, j by ĵ and k by ˆk. That is the reason for my peculiar choice of names for the matrix entries. a b = det î ĵ ˆk a 1 a 2 a 3 b 1 b 2 b 3 The above definition is good from the point of view of computing a b. Our first properties of the cross product lead up to a geometric definition of a b, which is better for interpreting a b. These properties of the cross product, which state that a b is a vector and then determine its direction and length, are as follows: 0. a, b are vectors in three dimensions and a b is a vector in three dimensions 1. a b a, b Proof: To checkthat a and a b areperpendicular, one just hasto check that the dot product a ( a b) = 0. The six terms in a ( a b) = a 1 (a 2 b 3 a 3 b 2 ) + a 2 (a 3 b 1 a 1 b 3 ) + a 3 (a 1 b 2 a 2 b 1 ) cancel pairwise. The computation showing that b ( a b) = 0 is similar. c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 10
11 2. a b = a b sinθ where θ is the angle between a and b = the area of the parallelogram with sides a and b Proof: This follows from a b 2 = a 2 b 2 ( a b) 2 = a 2 b 2 (1 cos 2 θ) which in turn is gotten by comparing a b 2 = (a 2 b 3 a 3 b 2 ) 2 +(a 3 b 1 a 1 b 3 ) 2 +(a 1 b 2 a 2 b 1 ) 2 = a 2 2b 2 3 2a 2 b 3 a 3 b 2 +a 2 3b 2 2 +a 2 3b 2 1 2a 3 b 1 a 1 b 3 +a 2 1b 2 3 +a 2 1b 2 2 2a 1 b 2 a 2 b 1 +a 2 2b 2 1 and a 2 b 2 ( a b) 2 = ( a 2 1 +a2 2 3)( +a2 b 2 1 +b 2 ) ( ) 2 2 +b2 3 a1 b 1 +a 2 b 2 +a 3 b 3 = a 2 1 b2 2 +a2 1 b2 3 +a2 2 b2 1 +a2 2 b2 3 +a2 3 b2 1 +a2 3 b2 2 ( ) 2a 1 b 1 a 2 b 2 +2a 1 b 1 a 3 b 3 +2a 2 b 2 a 3 b 3 To see that a b sinθ is the area of the parallelogram with sides a and b, just recall that the area of any parallelogram is given by the length of its base times its height. Think of a as the base of the parallelogram. Then a is the length of the base and b sinθ is the height. Thesepropertiesalmostdetermine a b. Property1forcesthe vector a btolie onthe lineperpendicular to the plane containing a and b. There are precisely two vectors on this line that have the length given by property 2. In the left figure of b θ a a b c b a a b b a d the two vectors are labeled c and d. Which of these two candidates is correct is determined by the right hand rule, which says that if you form your right hand into a fist with your fingers curling from a to b, then when you stick your thumb straight out from the fist, it points in the direction of a b. This is illustrated in the figure on the right above. The important special cases 3. î ĵ = ˆk, ĵ ˆk = î, ˆk î = ĵ all follow directly from the definition of the cross product and all obey the right hand rule. Combining properties 1, 2 and the right hand rule give the geometric definition of a b. 4. a b = a b sinθ ˆn where θ is the angle between a and b, ˆn = 1, ˆn a, b and ( a, b,ˆn) obey the right hand rule Outline of Proof: We have already seen that the right hand side has the correct length and, except possibly for a sign, direction. To check that the right hand rule holds in general, rotate your coordinate system around so that a points along the positive x axis and b lies in the xy plane with positive y component. That is a = αî and b = βî+γĵwith α,γ 0. Then a b = αî (βî+γĵ) = αβî î+αγî ĵ. The first term vanishes by property 2, because the angle θ between î and î is zero. So, by property 3, a b = αγˆk points along the positive z axis, which is consistent with the right hand rule. The analog of property 7 of the dot product follows immediately from property a b = 0 a = 0 or b = 0 or a b The remaining properties are all tools for helping do computations with cross products. 6. a b = b a 7. (s a) b = a (s b) = s( a b) c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 11
12 8. a ( b+ c) = a b+ a c 9. a ( b c) = ( a b) c 10. a ( b c) = ( c a) b ( b a) c WARNING: Take particular care with properties 6 and 10. They are counterintuitive and cause huge numbers of errors. In particular, for most a, b and c. For example a b b a a ( b c) ( a b) c î (î ĵ) = î ˆk = ˆk î = ĵ (î î) ĵ = 0 ĵ = 0 Example I.4 As an illustration of the properties of the dot and cross product, we now derive the formula for the volume of the parallelopiped with edges a = [a 1,a 2,a 3 ], b = [b 1,b 2,b 3 ], c = [c 1,c 2,c 3 ] that was mentioned in I.5. The volume of the parallelopiped is the area of its base times its height. The base is the b c a θ θ c b parallelogram with sides b and c. Its area is the length of its base, which is b, times its height, which is c sinθ. (Drop a perpendicular from the head of c to the line containing b). Here θ is the angle between b and c. So the area of the base is b c sinθ = b c, by property 2 of the cross product. To get the height of the paralleopiped, we drop a perpendicular from the head of a to the line that passes through the tail of a and is perpendicular to the base of the paralellopiped. In other words, from the head of a to the line that contains both the head and the tail of b c. So the height of the paralleopiped is a cosθ (I have included the absolute values because if the angle between b c and a happens to be greater than 90, the cosθ produced by taking the dot product of a and ( b c) will be negative). All together volume of parallelopiped = (area of base)(height) = b c a cosθ = a ( b c) = a 1 ( b c) 1 +a 2 ( b c) 2 +a 3 ( b c) 3 [ ] [ ] [ = a b2 b 3 b1 b 1det a c 2 c 2 det 3 b1 b +a 3 c 1 c 3 det 2 3 c 1 c 2] = det a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 12
13 Exercises for I.6 1) Compute (1,2,3) (4,5,6). 2) Show that a ( b c) = ( a b) c. 3) Show that a ( b c) = ( a c) b ( a b) c. 4) Derive a formula for ( a b) ( c d) that involves dot but not cross products. I.7 Application of Cross Products to Rotational Motion In most computations involving rotational motion, the cross product shows up in one form or another. This is one of the main applications of the cross product. Consider, for example, a rigid body which is rotating at a rate Ω radians per second about an axis whose direction is given by the unit vector â. Let P be any point on the body. Let s figure out its velocity. Pick any point on the axis of rotation and designate it as the origin of our coordinate system. Denote by r the vector from the origin to the point P. Let θ denote the angle between â and r. As time progresses the point P sweeps out a circle of radius R = r sinθ. In one 0 â θ r Ω v P second it travels along an arc that subtends an angle of Ω radians, which is the fraction Ω 2π of a full circle. The length of this arc is Ω 2π 2πR = ΩR = Ω r sinθ so P travels the distance Ω r sinθ in one second and its speed, which is also the length of its velocity vector, is Ω r sinθ. Now we just need to figure out the direction of the velocity vector. That is, the direction of motion of the point P. Imagine that both â and r lie in the plane of a piece of paper, as in the figure above. Then v points either straight into or straight out of the page and consequently is perpendicular to both â and r. To distinguish between the into the page and out of the page cases, let s impose the conventions that Ω > 0 and the axis of rotation â is chosen to obey the right hand rule, meaning that if the thumb of your right hand is pointing in the direction â, then your fingers are pointing in the direction of motion of the rigid body. Under these conventions, the velocity vector v obeys v = Ω r â sinθ v â, r (â, r, v) obey the right hand rule which is exactly the description of Ωâ r. It is conventional to define the angular velocity of a rigid body to be Ω = Ωâ. That is, the vector with length given by the rate of rotation and direction given by the axis of rotation of the rigid body. In terms of this angular velocity vector, the velocity of the point P is v = Ω r Exercises for I.7 1) A body rotates at an angular velocity of 10 rad/sec about the axis through (1,1, 1) and (2, 3,1). Find the velocity of the point (1,2,3) on the body. 2) Imagine a plate that lies in the xy plane and is rotating about the z axis. Let P be a point that is painted on this plate. Denote by r the distance from P to the origin, by θ(t) the angle at time t between the line from O to P and the x axis and by ( x(t),y(t) ) the coordinates of P at time t. Find x(t) and y(t) in terms of θ(t). Compute the velocity of P at time t by differentiating [x(t),y(t)]. Compute the velocity of P at time t by applying v = Ω r. c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 13
14 I.8 Equations of Lines in Two Dimensions A line in two dimensions can be specified by giving one point (x 0,y 0 ) on the line and one vector d = [d x,d y ] whose direction is parallel to the line. If (x,y) is any point on the line then the vector (x,y) d (x 0,y 0 ) [x x 0,y y 0 ], whose tail is at (x 0,y 0 ) and whose head is at (x,y), must be parallel to d and hence a scalar multiple of d. So [x x 0,y y 0 ] = t d or, writing out in components, x x 0 = td x y y 0 = td y These are called the parametric equations of the line, because they contain a free parameter, namely t. As t varies from to, the point (x 0 +td x,y 0 +td y ) runs from one end of the line to the other. It is easy to eliminate the parameter t from the equations. Just solve for t in the two equations t = x x0 d x t = y y0 d y Equating these two expressions for t gives x x 0 d x = y y0 d y which is called the symmetric equation for the line. In the event that the line is parallel to one of the axes, one of d x and d y is zero and we have to be a little careful to avoid division by zero. To do so, just multiply x x 0 = td x by d y, multiply y y 0 = td y by d x and subtract to give (x x 0 )d y (y y 0 )d x = 0 A second way to specify a line in two dimensions is to give one point (x 0,y 0 ) on the line and one vector n = [n x,n y ] whose direction is perpendicular to that of the line. If (x,y) is any point on the line then the (x,y) n (x 0,y 0 ) vector [x x 0,y y 0 ], whose tail is at (x 0,y 0 ) and whose head is at (x,y), must be perpendicular to n so that n [x x 0,y y 0 ] = 0 Writing out in components n x (x x 0 )+n y (y y 0 ) = 0 or n x x+n y y = n x x 0 +n y y 0 Observe that the coefficients n x,n y of x and y in the equation of the line are the components of a vector [n x,n y ] perpendicular to the line. This enables us to read off a vector perpendicular to any given line directly from the equation of the line. Such a vector is called a normal vector for the line. c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 14
15 Example I.2 Consider, for example, the line y = 3x+7. To rewrite this equation in the form n x x+n y y = n x x 0 +n y y 0 we have to move terms around so that x and y are on one side of the equation and 7 is on the other side: 3x y = 7. Then n x is the coefficient of x, namely 3, and n y is the coefficient of y, namely 1. One normal vector for y = 3x+7 is [3, 1]. To verify that [3, 1] really is perpendicular to the line, we can rewrite y = 3x+7 in the form n [x x 0,y y 0 ] = 0. Note that when (x,y) obeys y = 3x+7 and x = 0, we have y = 7. Thus (0,7) is one point on the line. 3x y = 7 [3, 1] [x,y] = 7 [3, 1] [x,y] = [3, 1] [0,7] [3, 1] ([x,y] [0,7]) = 0 [3, 1] [x 0,y 7] = 0 Now [x 0,y 7]is a vectorwhich has both head, namely (x,y), and tail, namely (0,7) on the line y = 3x+7. So [x 0,y 7] is a vector that is parallel to the line. The vanishing of the last dot product tells us that [3, 1] is perpendicular to [x 0,y 7] and hence to y = 3x+7. Of course, if [3, 1] is perpendicular to y = 3x+7, so is 5[3, 1] = [ 15,5]. In fact, if we first multiply the equation 3x y = 7 by 5 to get 15x+5y = 35 and then set n x and n y to the coefficients of x and y respectively, we get n = [ 15,5]. Example I.3 In this example, we find the point on the line y = 6 3x (call the line L) that is closest to (7,5). We ll start by sketching the line. To do so, we guess two points on L and then draw the line that passes through the two points. If (x,y) is on L and x = 0, then y = 6. So (0,6) is on L. If (x,y) is on L and y = 0, then x = 2. So (2,0) is on L. y L (0,6) P [3,1] N (7,5) (x,y) (2,0) x To find the point on L that is nearest to (7,5), we drop a perpendicular from (7,5) to L. The perpendicular hits L at the point we want, which we ll call P. Let s use N to denote the line which passes through (7,5) and which is perpedicular to L. Since L has the equation 3x + y = 6, one vector perpedicular to L, and hence parallel to N, is [3,1]. So if (x,y) is any point on N, the vector [x 7,y 5] must be of the form t[3,1]. So the parametric equations of N are [x 7,y 5] = t[3,1] or x = 7+3t, y = 5+t Now let (x,y) be the coordinates of P. Since P is on N, we have x = 7+3t, y = 5+t for some t. Since P is also on L, we also have 3x+y = 6. So and P is (1,3). Exercises for I.8 3(7+3t)+(5+t) = 6 10t+26 = 6 t = 2 = x = 7+3 ( 2) = 1, y = 5+( 2) = 3 c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 15
16 1) Use a projection to find the distance from the point ( 2,3) to the line 3x 4y = 4. 2) Let a, b and c be the vertices of a triangle. By definition, a median of a triangle is a straight line that passes through a vertex of the triangle and through the midpoint of the opposite side. a) Find the parametric equations of the three medians. b) Do the three medians meet at a common point? If so, which point? I.9 Equations of Planes in Three Dimensions Specifying one point (x 0,y 0,z 0 ) on a plane and a vector d parallel to the plane does not uniquely determine the plane, because it is free to rotate about d. On the other hand, giving one point on the plane d (x 0,y 0,z 0 ) n (x 0,y 0,z 0 ) (x,y,z) and one vector n = [n x,n y,n z ] whose direction is perpendicular to that of the plane does uniquely determine the plane. If (x,y,z) is any point on the line then the vector [x x 0,y y 0,z z 0 ], whose tail is at (x 0,y 0,z 0 ) and whose arrow is at (x,y,z), must be perpendicular to n so that Writing out in components n [x x 0,y y 0,z z 0 ] = 0 n x (x x 0 )+n y (y y 0 )+n z (z z 0 ) = 0 or n x x+n y y +n z z = n x x 0 +n y y 0 +n z z 0 Again, the coefficients n x,n y,n z of x, y and z in the equation of the plane are the components of a vector [n x,n y,n z ] perpendicular to the plane. Exercises for I.9 1) Find the equation of the plane containing the points (1,0,1), (1,1,0) and (0,1,1). 2) Find the equation of the sphere which has the two planes x+y+z = 3, x+y+z = 9 as tangent planes if the center of the sphere is on the planes 2x y = 0, 3x z = 0. 3) Find the equation ofthe plane that passesthrough the point ( 2,0,1)and through the line ofintersection of 2x+3y z = 0, x 4y+2z = 5. 4) What swrongwiththequestion Findtheequationoftheplanecontaining(1,2,3),(2,3,4)and(3,4,5).? 5) Find the distance from the point p to the plane n x = c. I.10 Equations of Lines in Three Dimensions Just as in two dimensions, a line in three dimensions can be specified by giving one point (x 0,y 0,z 0 ) on the line and one vector d = [d x,d y,d z ] whose direction is parallel to that of the line. If (x,y,z) is any point on the line then the vector [x x 0,y y 0,z z 0 ], whose tail is at (x 0,y 0,z 0 ) and whose arrow is at (x,y,z), must be parallel to d and hence a scalar multiple of d. Translating this statement into a vector equation [x x 0,y y 0,z z 0 ] = t d c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 16
17 or the three corresponding scalar equations x x 0 = td x y y 0 = td y z z 0 = td z again gives the parametric equations of the plane. Solving all three equations for the parameter t t = x x0 d x = y y0 d y = z z0 d z and erasing the t = again gives the symmetric equations for the line. Example I.4 The set of points (x,y,z) that obey x+y+z = 2 form a plane. The set of points (x,y,z) that obey x y = 0 form a second plane. The set of points (x,y,z) that obey both x+y +z = 2 and x y = 0 lie on the intersection of these two planes and hence form a line. We shall find the parametric equations for that line. To sketch x+y+z = 2 we observe that if any two of x,y,z are zero, then the third is 2. So all of (0,0,2), (0,2,0) and (2,0,0) are on x+y +z = 2. The plane x y = 0 contains all of the z axis, since (0,0,z) obeys x y = 0 for all z. (0,0,2) x y = 0 (2,0,0) x+y +z = 2 d (0,2,0) (1,1,0) Method 1. Each point on the line has a different value of z. We ll use z as the parameter. (We could just as well use x or y.) There is no law that requires us to use the parameter name t, but that s what we have done so far, so set t = z. If (x,y,z) is on the line then z = t and x+y +t = 2 x y = 0 The second equation forces y = x. Substituting this into the first equation gives So the parametric equations are 2x+t = 2 = x = y = 1 t 2 x = 1 t 2, y = 1 t 2, z = t or [x 1,y 1,z] = t[ 1 2, 1 2,1] Method 2. We first find one point on the line. There are lots of them. We ll find the point with z = 0. (We could just as well use z=123.4.) If (x,y,z) is on the line and z = 0, then x+y = 2 x y = 0 The second equation forces again y = x. Substituting this into the first equation gives 2x = 2 = x = y = 1 So (1,1,0) is on the line. Now we ll find a direction vector, d, for the line. Since the line is contained in the plane x+y + z = 2, any vector lying on the line, like d, is also completely contained in that plane. So d c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 17
18 must be perpendicular to the normal vector of x+y +z = 2, which is [1,1,1]. Similarly, since the line is contained in the plane x y = 0, any vector lying on the line, like d, is also completely contained in that plane. So d must be perpendicular to the normal vector of x y = 0, which is [1, 1,0]. So we may choose for d any vector which is perpendicual to both [1,1,1] and [1, 1,0], like, for example, d = [1, 1,0] [1,1,1] = det î = î ĵ+2ˆk ĵ ˆk = îdet[ ] ĵdet [ ] 1 0 +ˆkdet 1 1 [ ] We now have both a point on the line and a direction vector for the line, so, as usual, the parametric equations for the line are [x 1,y 1,z] = t[ 1, 1,2] or x = 1 t, y = 1 t, z = 2t Method 3. We ll find two points on the line. We have already found that (1,1,0) is on the line. From the picture, it looks like (0,0,2) is also on the line. This is indeed the case since (0,0,2) obeys both x+y+z = 2 and x y = 0. As both (1,1,0) and (0,0,2) are on the line, the vector with head at (1,1,0) and tail at (0,0,2), which is [1 0,1 0,0 2] = [1,1, 2], is a direction vector for the line. As (0,0,2) is a point on the line and [1,1, 2] is a direction vector for the line, the parametric equations for the line are [x 0,y 0,z 2] = t[1,1, 2] or x = t, y = t, z = 2 2t The parametric equations given by the three methods are different. That s just because we have really used different parameters in the three methods, even though we have always called the parameter t.to clarify the relation between the three answers, rename the parameter of method 1 to t 1, the parameter of method 2 to t 2 and the parameter of method 3 to t 3. The parametric eqautions then become Method 1: x = 1 t1 2 y = 1 t1 2 z = t 1 Method 2: x = 1 t 2 y = 1 t 2 z = 2t 2 Method 3: x = t 3 y = t 3 z = 2 2t 3 Substituting t 1 = 2t 2 into the Method 1 equations gives the Method 2 equations, and substituting t 3 = 1 t 2 into the Method 3 equations gives the Method 2 equations. Exercises for I.10 1) Find the equation of the line through (2, 1, 1) and parallel to each of the two planes x+y = 0 and x y+2z = 0. Express the equations of the line in vector and scalar parametric forms and in symmetric form. I.11 Worked Problems Questions 1) Describe the set of all points (x,y,z) in IR 3 that satisfy x 2 +y 2 +z 2 = 2x 4y +4. 2) Describe the set of all points (x,y,z) in IR 3 that satisfy x 2 +y 2 +z 2 < 2x 4y +4. 3) Compute the areas of the parallelograms determined by the following vectors. a) [ 3,1], [4,3] b) [4,2], [6,8] 4) Compute the volumes of the parallelopipeds determined by the following vectors. a) [4,1, 1], [ 1,5,2], [1,1,6] b) [ 2,1,2], [3,1,2], [0,2,5] c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 18
19 5) Determine the angle between the vectors a and b if a) a = [1,2], b = [3,4] b) a = [2,1,4], b = [4, 2,1] c) a = [1, 2,1], b = [3,1,0] 6) Determine whether the given pair of vectors is perpendicular a) [1,3,2], [2, 2,2] b) [ 3,1,7], [2, 1,1] c) [2,1,1], [ 1,4,2] 7) Determine all values of y for which the given vectors are perpendicular a) [2,4], [2,y] b) [4, 1], [y,y 2 ] c) [3,1,1], [2,5y,y 2 ] 8) Determine a number α such that [1,2,3] is perpendicular to [α,2,α]. 9) Let u = 2î+5ĵ and v = αî 2ĵ. Find α so that a) u v b) u v c) The angle between u and v is ) Find the angle between the diagonal of a cube and the diagonal of one of its faces. 11) Define a = [1,2,3], b = [4,10,6]. a) Find the component of b in the direction a. b) Find the projection of b on a. c) Find the projection of b perpendicular to a. 12) Consider the following statement: If a 0 and if a b = a c then b = c. If the statment is true, prove it. If the statement is false, give a counterexample. 13) Consider a cube such that each side has length s. Name, in order, the four vertices on the bottom of the cube A,B,C,D and the corresponding four vertices on the top of the cube A,B,C,D. a) Show that all edges of the tetrahedron A C BD have the same length. b) Let E be the center of the cube. Find the angle between EA and EC. 14) A prism has the six vertices A = (1,0,0) A = (5,0,1) B = (0,3,0) B = (4,3,1) C = (0,0,4) C = (4,0,5) a) Verify that three of the faces are parallelograms. Are they rectangular? b) Find the length of AA. c) Find the area of the triangle ABC. d) Find the volume of the prism. 15) The figure below represents a pin jointed network in equilibrium. The line ACD is horizontal. Each of AC, CD, BC and BD are 2m long. The only external force is a downward force of 10n applied at C. The support A is completely fixed, whereas C provides only vertical support. Determine the tensions N i in the five rods, using the sign convention that N i > 0 when rod number i is pulling on its ends, rather than pushing on them. B N 1 N 3 N 2 A N 4 C N 5 D 16) Let PQR be a triangle in IR 3. Find the work done in moving an object around the triangle when it is subject to a constant force F. c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 19
20 17) Calculate the following cross products. a) [1, 5,2] [ 2,1,5] b) [2, 3, 5] [4, 2,7] c) [ 1,0,1] [0,4,5] 18) Let p = [ 1,4,2], q = [3,1, 1], r = [2, 3, 1]. Check, by direct computation, that (a) p p = 0 (b) p q = q p (c) p (3 r) = 3( p r) (d) p ( q + r) = p q + p r (e) p ( q r) ( p q) r 19) Calculate the area of the triangle with vertices (0,0,0) (1,2,3) and (3,2,1). 20) Show that the area of the parallelogram spanned by the vectors a and b is a b. 21) Show that the volume of the parallelopiped spanned by the vectors a, b and c is a ( b c). 22) (Three dimensional Pythagorean Theorem) A solid body in space with exactly four vertices is called a tetrahedron. Let A, B, C and D be the areas of the four faces of a tetrahedron. Suppose that the three edges meeting at the vertex opposite the face of area D are perpendicular to each other. Show that D 2 = A 2 +B 2 +C 2. b A B C a c 23) (Three dimensional law of cosines) Let A, B, C and D be the areas of the four faces of a tetrahedron. Let α be the angle between the faces with areas B and C, β be the angle between the faces with areas A and C and γ be the angle between the faces with areas A and B. (By definition, the angle between two faces is the angle between the normal vectors to the faces.) Show that D 2 = A 2 +B 2 +C 2 2BCcosα 2ACcosβ 2ABcosγ 24) Consider the following statement: If a 0 and if a b = a c then b = c. If the statment is true, prove it. If the statement is false, give a counterexample. 25) Consider the following statement: The vector a ( b c) is of the form α b+β c for some real numbers α and β. If the statment is true, prove it. If the statement is false, give a counterexample. 26) What geometric conclusions can you draw from a ( b c) = [1,2,3]? 27) What geometric conclusions can you draw from a ( b c) = 0? 28) Find the vector parametric, scalar parametric and symmetric equations for the line containing the given point and with given direction. a) point (1, 2), direction [3, 2] c) point (5, 4), direction [2, 1] d) point ( 1, 3), direction [ 1, 2] 29) Find the vector parametric, scalar parametric and symmetric equations for the line containing the given point and with given normal. a) point (1,2), normal [3,2] c) point (5,4), normal [2, 1] d) point ( 1, 3), normal [ 1, 2] 30) Find a vector parametric equation for the line of intersection of the given planes. a) x 2z = 3 and y z = 5 c Joel Feldman All rights reserved. January 23, 2011 Vectors and Geometry 20
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